Analytical and low-order numerical models are very useful for studying friction behavior of rolling element machine components like ball bearings and ball screws. This is because they provide generalizable insights into friction behavior at much lower computational costs compared with high-order numerical models like finite element analysis (FEA). While analytical and low-order numerical models in the literature are mainly focused on ball-to-groove contact friction, experimental studies have shown that ball-to-ball contact friction is also very important. This is especially true for linear ball bearings/guideways and ball screws which, unlike rotary ball bearings, do not typically make use of caged balls to prevent ball-to-ball contact. Therefore, in this paper, low-order numerical models for ball-to-ball contact friction in linear ball bearings and ball screws are developed. Furthermore, an analytical model for ball-to-ball contact friction in four-point contact linear ball bearing is derived by making simplifications to its low-order numerical model. Compared with ball-to-ball friction predictions from FEA models developed in ansys, the proposed numerical models are shown in case studies to be accurate within 7%, while computing at least three orders of magnitude faster. Moreover, case studies are used to demonstrate how the developed models can be used in practice, e.g., for the mitigation of ball-to-ball contact friction in linear ball bearings and the prediction of friction variation during the operation of a ball screw.

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